Ultrafast magnetization and energy flow in the laser-induced dynamics of transition metal compounds

An electronic theory of the laser-induced ultrafast magnetization dynamics in transition-metal alloys is presented. A many-body model Hamiltonian is considered that incorporates the fundamental electronic hopping, local Coulomb interaction, and spin-orbit coupling (SOC) on the same footing. Exact time propagations are performed on a tetrahedral cluster, from which the time dependencies of the local spin moments, orbital occupations, and single-particle energies of homogeneous systems and binary alloys are obtained. The consequences of inhomogeneities in the laser absorption and in the SOC strengths are investigated giving emphasis to the nature of spin-density transfers between the alloy components. A local perspective on the optically induced spin transfer is proposed in terms of which two main steps emerge: a dominantly local optical excitation followed by hopping-driven spin-density transfers among the different alloy components. The conjoint action of the spin-orbit interactions at the origin of local spin flips and spin-to-orbital angular-momentum transfer and the interatomic hoppings responsible for charge, spin, and energy flows between different sublattices is demonstrated. Transient and steady-state dynamical regimes are identified that result in delays in the onset of the local demagnetizations and in ultrafast redistributions of the spin. The central role of electron delocalization and electronic hopping in the spin-density redistribution and demagnetization of the different alloy components is demonstrated.

Figure 1

Time dependence of the average spin magnetization $S_{iz}\left(t\right)$ of sublattice A (blue curves) and sublattice B (red curves) after an inhomogeneous laser pulse acting only on sublattice A (solid curves) and after a homogeneous absorption in both sublattices (dashed curves). The pulse is centered at $t=0$ and has a duration ${\tau }_p=1\mathrm{fs}$. The inset shows the tetrahedral cluster used for the exact time propagations.

Figure 2

Time dependence of (a) the local spin polarization $S_{iz}$, (b) the majority-spin $d$-electron (total electron) occupations $n_{id\uparrow }$ ($n_{iT}$), and (c) the local absorbed excitation energies $\mathrm{\Delta }{\mathcal{E}}_q\left(t\right)={\mathcal{E}}_q\left(t\right)-{\mathcal{E}}_q^0$ corresponding to the sublattices A (blue curves) and B (red curves). Results are given for an inhomogeneous laser absorption on sublattice A (solid curves) and a homogeneous laser absorption on both sublattices (dashed curves). The dotted curve in (c) shows the average absorbed energy $\overline{\mathrm{\Delta }\mathcal{E}}=(\mathrm{\Delta }{\mathcal{E}}_A+\mathrm{\Delta }{\mathcal{E}}_B)/2$ in the inhomogeneous case. The pulse is centered at $t=0$ and has a duration of ${\tau }_p=1\mathrm{fs}$ as illustrated in (a).

Figure 3

Time dependence of (a) the majority-spin $d$-electron (total electron) occupations $n_{id\uparrow }$ ($n_{iT}$) and (b) the local absorbed excitation energies $\mathrm{\Delta }{\mathcal{E}}_q\left(t\right)={\mathcal{E}}_q\left(t\right)-{\mathcal{E}}_q^0$ corresponding to the sublattices A (blue curves) and B (red curves). Results are given for an inhomogeneous laser absorption on sublattice A. The green curve in (b) shows the average absorbed energy $\overline{\mathrm{\Delta }\mathcal{E}}=(\mathrm{\Delta }{\mathcal{E}}_A+\mathrm{\Delta }{\mathcal{E}}_B)/2$. The pulse is centered at $t=0$ and has a duration of ${\tau }_p=20\mathrm{fs}$ as illustrated in (a).

Figure 4

Average spin magnetization $S_{iz}\left(t\right)$ at atoms A (blue curves) and B (red curves) as a function of time $t$, for ${\xi }_B=50$ meV and different spin-orbit coupling strengths ${\xi }_A$ at sublattice A. In all cases the same initial excited state obtained by laser excitation of the homogeneous system with ${\xi }_A={\xi }_B=50$ meV (dashed curves) is considered. The dotted curves are obtained by fitting the solution of the phenomenological rate equations (6) and (7) to the corresponding exact time propagations $S_{iz}\left(t\right)$. In the inset, the logarithm of the normalized magnetization change ${\sigma }_i\left(t\right)=[S_{iz}\left(t\right)-S_{iz}\left(\infty \right)]/[S_{iz}^0-S_{iz}\left(\infty \right)]$ is shown as a function of time. The time shifts $\mathrm{\Delta }$ between the onset of the demagnetization in sublattices A and B are indicated by the horizontal arrows, as obtained from Eq. (8): $\mathrm{\Delta }=8$ fs for ${\xi }_A=30$ meV and $\mathrm{\Delta }=-5$ fs for ${\xi }_A=70$ meV.

Figure 5

Average spin magnetization $S_{iz}\left(t\right)$ at atoms A (blue curves) and B (red curves) as a function of time $t$ for ${\xi }_A=100$ meV and ${\xi }_B=50$ meV in comparison with the homogeneous systems having $\xi =50$ and $\xi =100$ meV (dashed curves). In all cases the same initial excited state corresponding to the laser excitation of the homogeneous system with ${\xi }_A={\xi }_B=50$ meV is considered.

Figure 6

(a) Local demagnetization rate $k_A$ of sublattice A (blue open triangles) and time shift $\mathrm{\Delta }$ of the demagnetization of sublattice A with respect to sublattice B (magenta open circles). Results are given as a function of the spin-orbit coupling strength ${\xi }_A$ at the A atoms for ${\xi }_B=50$ meV. $k_A$ is obtained by fitting the solution of the rate equations (6) and (7) to the corresponding exact $S_z^A\left(t\right)$ and $S_z^B\left(t\right)$ using $k_B=0.021 {\mathrm{fs}}^{-1}$ and $k_{AB}=0.05 {\mathrm{fs}}^{-1}$. The time shift $\mathrm{\Delta }$ follows then from Eq. (8). (b) Sublattice demagnetization times ${\tau }_{\mathrm{dm}}^A$ (blue filled rhombus) and ${\tau }_{\mathrm{dm}}^B$ (red crosses). The lines connecting the points and the dashed straight line with $k_{\mathrm{A}}\simeq 5\times {10}^{-4} {\mathrm{fs}}^{-1}$ ${\mathrm{meV}}^{-1}{\xi }_A$ are a guide to the eyes.

Figure 7

Time dependence of the local spin magnetization $S_{iz}\left(t\right)$ at the A atoms (blue curves) and at the B atoms (red curves) after excitation with a 1 fs laser pulse (homogeneous field). Results are given for a binary alloy having ${\xi }_A=0$ and ${\xi }_B=50$ meV (solid curves) and for the homogeneous system having ${\xi }_A={\xi }_B=50\mathrm{meV}$ (dashed curves). The dotted curves are obtained by fitting the solution of the phenomenological rate equations (6) and (7) to the exact results. In the inset the logarithm of the normalized magnetization changes ${\sigma }_i\left(t\right)$ are shown. Here the horizontal arrow indicates the delay $\mathrm{\Delta }$ involved in the onset of the demagnetization of sublattice A, as obtained from Eq. (8).

Figure 8

Time dependence of the local spin moments $S_z^A$ (blue curves) and $S_z^B$ (red curves) in a laser-excited binary alloy having ${\xi }_A=0, {\xi }_B=50$ meV and representative values of the scaled hopping integrals $t_{ij}=\alpha t_{ij}^0$ between the sublattices A and B. The solid curves are the result of exact time propagations, while the dotted curves are obtained by fitting the phenomenological rate equations (6) and (7). The insets show the logarithm of the normalized magnetization change ${\sigma }_i\left(t\right)=[S_{iz}\left(t\right)-S_{iz}\left(\infty \right)]/[S_{iz}^0-S_{iz}\left(\infty \right)]$. The indicated time delays $\mathrm{\Delta }$ have been determined using Eq. (8).

Figure 9

(a) Time shift $\mathrm{\Delta }$ (magenta open circles) of the demagnetization of sublattice A with respect to sublattice B, interlattice spin-transfer rate $k_{AB}$ (purple asterisks) and local demagnetization rate $k_B$ (orange plus signs) as functions of the relative strength $\alpha =t_{ij}/t_{ij}^0$ of the hybridizations between the sublattices A and B. The hopping integrals between atoms of the same kind remain in all cases equal to the reference band-structure values $t_{ij}^0$. The results are derived from the fits to the exact time dependencies of $S_z^A\left(t\right)$ and $S_z^B\left(t\right)$ using the rate equations (6) and (7), which are shown in Fig. 8 [see also Eq. (8)]. (b) Sublattice demagnetization times ${\tau }_{\mathrm{dm}}^A$ (blue filled rhombus) and ${\tau }_{\mathrm{dm}}^B$ (red crosses). The lines connecting the points and the dashed straight line with $k_{\mathrm{AB}}\simeq {\left(16.7\mathrm{fs}\right)}^{-1}\alpha$ are a guide to the eyes.